3. If Ais a symmetric matrix and λiis an eigenvalue of multiplicity ri,then
there are rilinearly independent eigenvectors.
4. An n×n square matrix is similar to a diagonal matrix if it has n-
independent eigenvectors.
5. The set of all eigenvalues of Ais called the spectrum of the matrix A.
6. The largest (in absolute value) eigenvalue of the matrix Ais called the
spectral radius of A.
7. If Ais a real symmetric matrix, then all eigenvalues are real.
8. If Ais a real skew symmetric matrix, then all eigenvalues are imaginary.
9. If a=max |aij|and λis an eigenvalue of A, then |λ|≤ na.
10. An eigenvalue of Amust lie within one of ncircular disks whose centers
are aii, i = 1, 2 ,.. ., n, and whose radii are
ri=
∑n
jj=1=i
|aij|;
that is, the centers of these disks are determined by the elements along
the main diagonal of A, and the radius of the disk with center at aii is
obtained by deleting aii from the ith row and then summing the absolute
value of the remaining elements in the ith row.
11. The eigenvalues of a real matrix Asatisfy:
(a)
∑n
i=1
λi=
∑n
i=1
aii =Trace (A)
(b)
∏n
i=1
λi=λ 1 λ 2 ···λn= det (A)
(c)
∑n
i=1
λ^2 i≤
∑n
i=1
∑n
j=1
a^2 ij
Example 10-26. For the matrix A=
[ 5
4 −
√ 3
4
−
√ 3
4
7
4
]
find matrices Qand Q−^1
such that Q−^1 AQ is a diagonal matrix.
Solution: Calculate the eigenvalues λ 1 , λ 2 and eigenvectors X 1 , X 2 of the given matrix
Aand show
